If you are getting this message "You are not authorized to access table ‘Table label’ (TableName). Contact your system administrator." when trying to use the Table browser and you are a SysAdmin in AX, which is rather interesting, this is one possible thing you are overlooking:
make sure the table you are trying to browse is not having the property Visible set to False. If it does, just set it to True for the time you want to browse the table's content and it should (will) work.